Normalized defining polynomial
\( x^{16} - 4 x^{15} + 12 x^{14} - 21 x^{13} + 19 x^{12} - 37 x^{11} + 48 x^{10} - 63 x^{9} + 70 x^{8} - 45 x^{7} + 99 x^{6} - 38 x^{5} + 12 x^{4} - 38 x^{3} + 8 x^{2} + x + 1 \)
Invariants
| Degree: | $16$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[4, 6]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(6979237056235696902903125=5^{5}\cdot 29^{8}\cdot 109^{2}\cdot 613^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $35.71$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $5, 29, 109, 613$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{78} a^{14} - \frac{11}{78} a^{13} + \frac{11}{39} a^{11} - \frac{3}{13} a^{10} + \frac{1}{26} a^{9} - \frac{3}{26} a^{8} - \frac{11}{26} a^{7} + \frac{5}{39} a^{6} - \frac{25}{78} a^{5} + \frac{4}{39} a^{4} - \frac{7}{39} a^{3} - \frac{17}{78} a^{2} + \frac{1}{78} a + \frac{16}{39}$, $\frac{1}{398391630} a^{15} + \frac{7613}{28456545} a^{14} + \frac{8805983}{132797210} a^{13} - \frac{37033336}{199195815} a^{12} - \frac{23492918}{66398605} a^{11} + \frac{6618579}{26559442} a^{10} + \frac{1630489}{9485515} a^{9} - \frac{20642}{1897103} a^{8} + \frac{3385403}{11382618} a^{7} - \frac{2147021}{79678326} a^{6} + \frac{173163329}{398391630} a^{5} - \frac{6608377}{199195815} a^{4} - \frac{43891247}{398391630} a^{3} - \frac{2365235}{5691309} a^{2} - \frac{58117177}{398391630} a - \frac{10789721}{66398605}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $9$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -1 \) (order $2$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 2569068.46818 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 8192 |
| The 104 conjugacy class representatives for t16n1702 are not computed |
| Character table for t16n1702 is not computed |
Intermediate fields
| \(\Q(\sqrt{29}) \), 4.4.4205.1, 8.4.1181459864425.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/LocalNumberField/2.8.0.1}{8} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ | R | ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ | $16$ | ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ | R | ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ | $16$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ | $16$ | ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| $5$ | 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.4.3.4 | $x^{4} + 40$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 5.8.0.1 | $x^{8} + x^{2} - 2 x + 3$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| $29$ | 29.4.2.1 | $x^{4} + 145 x^{2} + 7569$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 29.4.2.1 | $x^{4} + 145 x^{2} + 7569$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 29.8.4.1 | $x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $109$ | $\Q_{109}$ | $x + 6$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{109}$ | $x + 6$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{109}$ | $x + 6$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{109}$ | $x + 6$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{109}$ | $x + 6$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{109}$ | $x + 6$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 109.2.0.1 | $x^{2} - x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 109.4.0.1 | $x^{4} - x + 30$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 109.4.2.1 | $x^{4} + 1199 x^{2} + 427716$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 613 | Data not computed | ||||||